New method for shadow calculations: Application to parametrized axisymmetric black holes

The first law of black hole mechanics (in the form derived by Wald) is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism-invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter, and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. One would expect that Wald's methods, applied to a Lagrangian Einstein-perfect fluid formulation, would convert these terms to surface integrals. However, because the fields appearing in the Lagrangian of a gravitating perfect fluid are typically nonstationary (even in a stationary black-hole--perfect-fluid spacetime) a direct application of these methods generally yields restricted results. We therefore first approach the problem of incorporating general nonstationary matter fields into Wald's analysis, and derive a first-law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter, and Hawking when the theory is general relativity coupled to a perfect fluid. We then turn to consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis, assuming that both the metric and fluid fields are stationary and axisymmetric in the black hole spacetime. The first law we derive contains only surface integrals at the black hole horizon and spatial infinity, but the assumptions of stationarity and axisymmetry of the fluid fields make this relation much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter, and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.

Einstein’s General theory of relativity (GR) successfully describes gravity. Although GR has been accurately tested in weak gravitational fields, it remains largely untested in the general strong field cases. One of the most fundamental predictions of GR is the existence of black holes (BHs). After the recent direct detection of gravitational waves by LIGO, there is now near conclusive evidence for the existence of stellar-mass BHs. In spite of this exciting discovery, there is not yet direct evidence of the existence of BHs using astronomical observations in the electromagnetic spectrum. Are BHs observable astrophysical objects? Does GR hold in its most extreme limit or are alternatives needed? The prime target to address these fundamental questions is in the center of our own Milky Way, which hosts the closest and best-constrained supermassive BH candidate in the universe, Sagittarius A* (Sgr A*). Three different types of experiments hold the promise to test GR in a strong-field regime using observations of Sgr A* with new-generation instruments. The first experiment consists of making a standard astronomical image of the synchrotron emission from the relativistic plasma accreting onto Sgr A*. This emission forms a “shadow” around the event horizon cast against the background, whose predicted size ([Formula: see text]as) can now be resolved by upcoming very long baseline radio interferometry experiments at mm-waves such as the event horizon telescope (EHT). The second experiment aims to monitor stars orbiting Sgr A* with the next-generation near-infrared (NIR) interferometer GRAVITY at the very large telescope (VLT). The third experiment aims to detect and study a radio pulsar in tight orbit about Sgr A* using radio telescopes (including the Atacama large millimeter array or ALMA). The BlackHoleCam project exploits the synergy between these three different techniques and contributes directly to them at different levels. These efforts will eventually enable us to measure fundamental BH parameters (mass, spin, and quadrupole moment) with sufficiently high precision to provide fundamental tests of GR (e.g. testing the no-hair theorem) and probe the spacetime around a BH in any metric theory of gravity. Here, we review our current knowledge of the physical properties of Sgr A* as well as the current status of such experimental efforts towards imaging the event horizon, measuring stellar orbits, and timing pulsars around Sgr A*. We conclude that the Galactic center provides a unique fundamental-physics laboratory for experimental tests of BH accretion and theories of gravity in their most extreme limits.

Following previous work of ours in spherical symmetry, we here propose a new\nparametric framework to describe the spacetime of axisymmetric black holes in\ngeneric metric theories of gravity. In this case, the metric components are\nfunctions of both the radial and the polar angular coordinates, forcing a\ndouble expansion to obtain a generic axisymmetric metric expression. In\nparticular, we use a continued-fraction expansion in terms of a compactified\nradial coordinate to express the radial dependence, while we exploit a Taylor\nexpansion in terms of the cosine of the polar angle for the polar dependence.\nThese choices lead to a superior convergence in the radial direction and to an\nexact limit on the equatorial plane. As a validation of our approach, we build\nparametrized representations of Kerr, rotating dilaton, and\nEinstein-dilaton-Gauss-Bonnet black holes. The match is already very good at\nlowest order in the expansion and improves as new orders are added. We expect a\nsimilar behavior for any stationary and axisymmetric black-hole metric.\n

The Rezzolla-Zhidenko (RZ) framework provides an efficient approach to\ncharacterize spherically symmetric black-hole spacetimes in arbitrary metric\ntheories of gravity using a small number of variables [L. Rezzolla and A.\nZhidenko, Phys. Rev. D. 90, 084009 (2014)]. These variables can be obtained in\nprinciple from near-horizon measurements of various astrophysical processes,\nthus potentially enabling efficient tests of both black-hole properties and the\ntheory of general relativity in the strong-field regime. Here, we extend this\nframework to allow for the parametrization of arbitrary asymptotically-flat,\nspherically symmetric metrics and introduce the notion of a 11-dimensional\n(11D) parametrization space $\\Pi$, on which each solution can be visualised as\na curve or surface. An $\\mathscr{L}^2$ norm on this space is used to measure\nthe deviation of a particular compact object solution from the Schwarzschild\nblack-hole solution. We calculate various observables, related to particle and\nphoton orbits, within this framework and demonstrate that the relative errors\nwe obtain are low (about $10^{-6}$). In particular, we obtain the innermost\nstable circular orbit (ISCO) frequency, the unstable photon-orbit impact\nparameter (shadow radius), the entire orbital angular speed profile for\ncircular Kepler observers and the entire lensing deflection angle curve for\nvarious types of compact objects, including non-singular and singular black\nholes, boson stars and naked singularities, from various theories of gravity.\nFinally, we provide in a tabular form the first 11 coefficients of the\nfourth-order RZ parameterization needed to describe a variety of commonly used\nblack-hole spacetimes. When comparing with the first-order RZ parameterization\nof astrophysical observables such as the ISCO frequency, the coefficients\nprovided here increase the accuracy of two orders of magnitude or more.\n

This manuscript gathers and reviews part of our work focusing on the exploration of modified theories of gravity known as degenerate higher order scalar-tensor (DHOST) theories. It focuses on the construction of exact solutions describing both black holes and radiative spacetimes. After motivating the need for alternatives theories of gravity beyond general relativity, we discuss in more details the long terms objectives of this research program. The first one is to characterize both the theory and some sectors of the solution space of DHOST gravity. The second one is to provide concrete and exact solutions of the DHOST field equations describing compact objects, in particular black holes, that can be used to confront DHOST theories to current and future observations in the strong field regime. A key tool towards these two objectives is the concept of disformal field redefinition (DFR) which plays a central role in this exploration. We start be reviewing the structure of the DHOST theory space, the notion of degeneracy conditions and the stability of these degeneracy classes under DFR. Then we review several key notions related to stationary and axi-symmetric black holes, and in particular the no-hair theorems derived in GR and in its scalar-tensor extensions. The rest of the sections are devoted to a review of the disformal solution generating map, the subtle role of matter coupling and how it can be used to construct new hairy black holes solutions. The case of spherically symmetric solutions, axi-symmetric but non-rotating solutions, and finally rotating solutions are discussed, underlining the advantages and the limitations of this approach. A brief review of the rotating black holes solutions found so far in this context is followed by the detailed description of the disformed Kerr black hole. We further comment on on-going efforts to construct rotating black hole solutions mimicking the closest the Kerr geometry. Then, we discuss how DFR affects the algebraic properties of a gravitational field and in particular its Petrov type. This provides a first systematic characterization of this effect, paving the road for constructing new solutions with a fixed Petrov type. Finally, we review more recent works aiming at characterizing the effect of a DFR on non-linear radiative geometries. We derive the general conditions for the generation of disformal tensorial gravitational wave and we study in detail a concrete example in DHOST gravity. While most of the material presented here is a re-organized and augmented version of our published works, we have included new results and also new proposals to construct phenomenologically interesting solutions.

One of the most interesting astronomical objects is the Galactic Center. It is a subject of intensive astronomical observations in different spectral bands in recent years. We concentrate our discussion on a theoretical analysis of observational data of bright stars in the IR-band obtained with large telescopes. We also discuss the importance of VLBI observations of bright structures which could characterize the shadow at the Galactic Center. If we adopt general relativity (GR), there are a number of theoretical models for the Galactic Center, such as a cluster of neutron stars, boson stars, neutrino balls, etc. Some of these models were rejected or the range of their parameters is significantly constrained with consequent observations and theoretical analysis. In recent years, a number of alternative theories of gravity have been proposed because there are dark matter (DM) and dark energy (DE) problems. An alternative theory of gravity may be considered as one possible solution for such problems. Some of these theories have black hole solutions, while other theories have no such solutions. There are attempts to describe the Galactic Center with alternative theories of gravity and in this case one can constrain parameters of such theories with observational data for the Galactic Center. In particular, theories of massive gravity are intensively developing and theorists have overcome pathologies presented in the initial versions of these theories. In theories of massive gravity, a graviton is massive in contrast with GR where a graviton is massless. Now these theories are considered as an alternative to GR. For example, the LIGO–Virgo collaboration obtained the graviton mass constraint of about [Formula: see text] eV in their first publication about the discovery of the first gravitational wave detection event that resulted of the merger of two massive black holes. Surprisingly, one could obtain a consistent and comparable constraint of graviton mass at a level around [Formula: see text][Formula: see text]eV from the analysis of observational data on the trajectory of the star S2 near the Galactic Center. Therefore, observations of bright stars with existing and forthcoming telescopes such as the European extremely large telescope (E-ELT) and the thirty meter telescope (TMT) are extremely useful for investigating the structure of the Galactic Center in the framework of GR, but these observations also give a tool to confirm, rule out or constrain alternative theories of gravity. As we noted earlier, VLBI observations with current and forthcoming global networks (like the Event Horizon Telescope) are used to check the hypothesis about the presence of a supermassive black hole at the Galactic Center.

This paper is written mostly in an overview style in its nature thus avoiding many equations and computations, which casual readers do not necessarily understand. Paper investigates and compares side by side in detail assumptions with their logical consequences and resulting internal inconsistencies in both; the General Relativity Theory and the Metric Theory of Gravity. It is found that the GRT has many such internal inconsistencies, which have to be corrected by unusual and difficult to believe assumptions that are not backed up by a typical experience one encounters in a real life, while the MTG avoids such problems. For the readers who are interested in proofs of discussed findings the paper provides internet links to papers where such proofs are available. The key differences between the GRT and MTG theories are: the gravitational mass dependence on velocity, nature of the “empty” space, the finite or infinite size of the Universe, the existence of Black Holes (BH) with their Event Horizons (EH), the creation of Universe by the Big Bang (BB), and the relation between the Cosmic Microwave Background Radiation (CMBR) temperature, and the Hubble constant that characterizes the velocity of receding Galaxies.   

We propose a new parametric framework to describe in generic metric theories of gravity the spacetime of spherically symmetric and slowly rotating black holes. In contrast to similar approaches proposed so far, we do not use a Taylor expansion in powers of M/r, where M and r are the mass of the black hole and a generic radial coordinate, respectively. Rather, we use a continued-fraction expansion in terms of a compactified radial coordinate. This choice leads to superior convergence properties and allows us to approximate a number of known metric theories with a much smaller set of coefficients. The measure of these coefficients via observations of near-horizon processes can be used to effectively constrain and compare arbitrary metric theories of gravity. Although our attention is here focussed on spherically symmetric black holes, we also discuss how our approach could be extended to rotating black holes.

In this work we study whether parametrized spherically symmetric black hole solutions in metric theories of gravity can appear to be isospectral when studying perturbations. From a theory agnostic point of view, the test scalar field wave equation is the ideal starting point to approach the quasinormal mode spectrum in alternative black hole solutions. We use a parametrization for the metric proposed by Rezzolla and Zhidenko, as well as the higher order WKB method in the determination of the quasinormal mode spectra. We look for possible degeneracies in a tractable subset of the parameter space with respect to the Schwarzschild quasinormal modes. Considering the frequencies and damping times of the expected observationally most relevant quasinormal modes, we find such degeneracies. We explicitly demonstrate that the leading Schwarzschild quasinormal modes can be approximated by alternative black hole solutions when their mass is treated as free parameter. In practice, we conclude that the mass has to be known with extremely high precision in order to restrict the leading terms in the metric expansion to currently known limits coming from the PPN expansion. Possible limitations of using the quasinormal mode ringdown to investigate black hole space-times are discussed.

We review the black hole (BH) solutions of the ghost-free massive gravity theory and its bimetric extension, and outline the main results on the stability of these solutions against small perturbations. Massive (bi)-gravity accommodates exact BH solutions, analogous to those of general relativity (GR). In addition to these solutions, hairy BHs—solutions with no correspondent in GR—have been found numerically, whose existence is a natural consequence of the absence of Birkhoff’s theorem in these theories. The existence of extra propagating degrees of freedom, makes the stability properties of these BHs richer and more complex than those of GR. In particular, the bi-Schwarzschild BH exhibits an unstable spherically symmetric mode, while the bi-Kerr geometry is also generically unstable, both against the spherical mode and against superradiant instabilities. If astrophysical BHs are described by these solutions, the superradiant instability of the Kerr solution imposes stringent bounds on the graviton mass.

The Event Horizon Telescope, a global submillimeter wavelength very long baseline interferometry array, produced the first image of supermassive black hole M87* showing a ring of diameter θ d = 42 ± 3 μas, inferred a black hole mass of M = (6.5 ± 0.7) × 109 M ⊙, and allowed us to investigate the nature of strong-field gravity. The observed image is consistent with the shadow of a Kerr black hole, which according to the Kerr hypothesis describes the background spacetimes of all astrophysical black holes. The hypothesis, a strong-field prediction of general relativity, may be violated in the modified theories of gravity that admit non-Kerr black holes. Here, we use the black hole shadow to investigate the constraints when rotating regular black holes (non-Kerr) can be considered as astrophysical black hole candidates, paying attention to three leading regular black hole models with additional parameters g related to nonlinear electrodynamics charge. Our interesting results based on the systematic bias analysis are that rotating regular black holes shadows may or may not capture Kerr black hole shadows, depending on the values of the parameter g. Indeed, the shadows of Bardeen black holes (g ≲ 0.26M), Hayward black holes (g ≲ 0.65M) and non-singular black holes (g ≲ 0.25M) are indistinguishable from Kerr black hole shadows within the current observational uncertainties, and thereby they can be strong viable candidates for the astrophysical black holes. Whereas Bardeen black holes (g ≤ 0.30182M), Hayward black holes (g ≤ 0.73627M), and non-singular black holes (g ≤ 0.30461M), within the 1σ region for θ d = 39 μas, are consistent with the observed angular diameter of M87*.

Accretion of magnetized gas on compact astrophysical objects such as black holes (BHs) has been successfully modeled using general relativistic magnetohydrodynamic (GRMHD) simulations. These simulations have largely been performed in the Kerr metric, which describes the spacetime of a vacuum and stationary spinning BH in general relativity (GR). The simulations have revealed important clues to the physics of accretion flows and jets near the BH event horizon and have been used to interpret recent Event Horizon Telescope images of the supermassive BHs M87* and Sgr A*. The GRMHD simulations require the spacetime metric to be given in horizon-penetrating coordinates such that all metric coefficients are regular at the event horizon. Only a few metrics, notably the Kerr metric and its electrically charged spinning analog, the Kerr–Newman metric, are currently available in such coordinates. We report here horizon-penetrating forms of a large class of stationary, axisymmetric, spinning metrics. These can be used to carry out GRMHD simulations of accretion on spinning, nonvacuum BHs and non-BHs within GR, as well as accretion on spinning objects described by non-GR metric theories of gravity.

The successes of f(R) gravitational theory as a logical extension of Einstein’s theory of general relativity (GR) encourage us to delve deep into this theory and continue our study to attempt to derive an extension of the Schwarzschild black hole (BH) solution. In this study, in order to solve the output nonlinear differential equation, we closed the form of the system by assuming the derivative of f(R) with respect to the scalar curvature R to have the form F(r)=df(R(r))dR(r)=1-αr4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(r)=\\frac{\ extrm{d}f(R(r))}{\ extrm{d}R(r)}=1- \\frac{\\alpha }{r^4}$$\\end{document}, where α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a dimensional constant. Our study shows that when α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document}, we obtain the Schwarzschild BH solution of GR assuming some constraints on the constant of integration, and if these constraints are bounded, we obtain the anti-de Sitter (AdS)/de Sitter (dS) spacetime. For the general case, i.e., when α≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \ e 0$$\\end{document}, we obtain a BH solution that tends asymptotically to AdS/dS spacetime. Moreover, we derive the timelike and null particle geodesics of the BH solution studied in this article. The equation of motion and effective potential of test particles are calculated to study the stability of radial orbits (trajectories). The energy and angular momentum are calculated to study the circular motion and stability of circular orbits. We also derive the stability condition using the geodesic deviation. Moreover, we discuss the physics of the output BH solutions through calculation of the thermodynamic quantities including entropy, the Hawking temperature, and Gibbs free energy. Finally, we check the validity of the first law of thermodynamics applied to the BH of this study. Although we can derive a Schwarzschild black hole solution in the lower order of f(R), specifically when f(R)=R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R)=R$$\\end{document}, where the gravitational mass is generated from the source of gravity, we demonstrate that in the higher orders of f(R), when f(R)≠R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(R)\ e R$$\\end{document}, the source of gravity is attributed primarily to higher-order corrections, and the source of gravity that was originally derived from the Schwarzschild black hole has ceased to be dominant.

Recent observations suggest that General Relativity (GR) faces challenges in fully explaining phenomena in regimes of strong gravitational fields. A promising alternative is the f(R) theory of gravity, where R denotes the Ricci scalar. This modified theory aims to address the limitations observed in standard GR. In this study, we derive a black hole (BH) solution without introducing nonlinear electromagnetic fields or imposing specific constraints on R or the functional form of f(R) gravity. The BH solution obtained here is different from the classical Schwarzschild solution in GR and, under certain conditions, reduces to the Schwarzschild (A)dS solution. This BH is characterized by the gravitational mass of the system and an additional parameter, which distinguishes it from GR BHs, particularly in the asymptotic regime. We show that the curvature invariants of this solution remain well defined at both small and large values of r. Furthermore, we analyze their thermodynamic properties, demonstrating consistency with established principles such as Hawking radiation, entropy, and quasi-local energy. This analysis supports their viability as alternative models to classical GR BHs.

Black Holes and the Supermassive Compact Object at the Galactic Center: Multi-arts of Thought and Nature